Solution.

We give two solutions.

Solution 1 (linear combination)

Since we know the values of $T$ on the basis vectors $\mathbf{v}_1, \mathbf{v}_2$, if we express the vector $\mathbf{x}$ as a linear combination of $\mathbf{v}_1, \mathbf{v}_2$, we can find $F(\mathbf{x})$ by the linearity of the linear transformation $T$.

Solution 2 (Matrix representation)

In the second solution, we use the matrix representation for the linear transformation $T$.
Let $A$ be the matrix of $T$ with respect to the standard basis $\{\begin{bmatrix}
1 \\
0
\end{bmatrix}, \begin{bmatrix}
0 \\
1
\end{bmatrix}\}$ of $\R^2$.
Thus, we have $T(\mathbf{x})=A\mathbf{x}$ by definition.